When you use the Open File function, you can specify read only as the open mode. If you can do that when the other program is doing a write, then you have to deal with knowing when the other program has finished doing it's write. If you don't have any direct way to communicate with the other program, knowing that is probably going to be a little tricky. Maybe you can read the number of lines in the file to determine whether the file is complete. If the other program opens the file exclusively, then you won't be able to open it until the other program is complete. You'll get an error back from the Open File function and you can keep trying until the error goes away and then do the read.
Of course, the most effecient way would be for you to communicate directly with the instrument using LabVIEW and not use an external program. What kind of instrument is it? How is it connected to the computer (serial, GPIB, etc.)? Have you tried to find a LabVIEW driver for it?